Connecting the Concepts (8.1-8.4)
Rational Expressions
The operations sign should be noted when combining rational expressions since the process of adding and subtracting rational
expressions is significantly different from multiplying and dividing them.
Need Common
Denominator?
Operation
Multiplication
No
Division
No
Do’s
2
2
Multiply: m  m  6  2m  5m  3
2m  4
m3
m 2  m  6 2 m 2  5m  3

2m  4
m3
 m  2   m  3  2m  3 m  1

2  m  2
 m  3
 2m  3 m  1
Procedure
-Factor all numerators and
denominators and simplify, if
possible.
-Write the factored form of the
product.
-Multiply the first rational
expression with reciprocal of the
second rational expression. (going
back to multiplication)
-Factor all numerators and
denominators and then simplify, if
possible.
-Write the factored form of the
product
Tips & Cautions
-Do not carry out the multiplication at
the beginning.
-Do not multiply the remaining factors
after simplifying but write the product
in factored form. Leave the final
expression in factored form.
-Begin by rewriting as a multiplication
before factoring, simplifying.
Don’ts
m 2  m  6 2m 2  5m  3

2m  4
m3
2m 4  5m 3  3m 2  2m 3  5m 2  3m  12m 2  30m  18

2m 2  6m  4m  12
2m 4  7m 3  4m 2  27 m  18

2m 2  2m  12
2
It is not necessary to carry
out these multiplications.
1
Need Common
Denominator?
Operation
Addition
Yes
Subtraction
Yes
Procedure
-If necessary, write equivalent
rational expressions with least
common denominator.
-Add numerators and keep common
denominators.
-Simplify, if possible.
-If necessary, write equivalent
rational expressions with least
common denominator.
-Subtract the numerators and
keep common denominators.
-Simplify, if possible.
Do’s
x
x  5x  6

2







x  3x
5

x2  2x  3 x  3
2
2

2
Factoring the denominators to find the LCD. LCD is
x
-Do not simplify after writing with the
LCD. Instead, simplify, if possible,
after subtracting the numerators.
- Do not divide out terms.
-Use parentheses around the numerator
being subtracted.
Don’ts
x  3x  2
 x  2  x  3   x  1 x  2 
 x  1
 x  3
x
2



 x  2 x  3  x  1  x  1 x  2  x  3
x
-Do not simplify after writing with the
LCD. Instead, simplify, if possible,
after adding the numerators.
-Do not divide out terms.
2
2
x
Tips & Cautions

 x  2   x  3  x  1 .
Writing equivallent expression with the LCD by myltiplying a form of 1.
2x  6


x  x  3
 x  3 x  1

5
 x  3
x 5
 x  3 x  1
 x  2 x  3 x  1  x  2 x  3 x  1
x
2

 x  2x  6

 x  2 x  3 x  1
x
2
Subtracting numerators. Don't forget the parenthesis!
 x  2x  6
 x  2 x  3 x  1
x
2
x6
 x  2 x  3 x  1
 x  2   x  3
 x  2  x  3 x  1
 x  3

 x  3 x  1
Combining like terms in the numerator.
Simplifying by factoring and removing a factor equal to 1.
4 z  9 3z  8

4z
3z
4 z  9   3z  8

4 z  3z
4 z  9  3z  8

z
z 1

z
4 z  9 3z  8

4z
3z
4 z  9 3z  8


4z
3z
 9    8 
 9  8
 1
2
Math 31
(Activity # ____)
Your Name: ___________________ Team Member #1__________________
Team Member #2.______________ Team Member #3__________________
Directions: Work collaboratively as a team to complete this activity.
Look at the errors of each of the following two problems present on page 2. Work as a team
to find the right way to approach the problem and present the solution step-by-step in an
organized fashion.
4 z  9 3z  8
x 2  3x
5

2)
1) 2

4z
3z
x  2x  3 x  3
Now try the following problems:
3) Multiply:
m 2  m  6 2m 2  5m  3

2m  4
m3
4) Subtract:
x
x2  5x  6

2
x 2  3x  2
3
Mixed Review: Perform the indicated operation and, if possible, simplify.
8
5
 2
3
9t
6t
6)
8
5
 2
3
9t 6t
7)
a3 a3

15a
3a 2
8)
a3 a3

15a
3a 2
9)
3
2

x4 4 x
10)
x 2  16
x2

x 2  x x 2  5x  4
11)
3u 2  3 4u  4

4
3
12)
3u 2  3 4u  4

4
3
5)
Answers:
1)
4)
7)
10)
x 3  11 x 2  x  15
x  3x  1x  3
 x  3
x  3x  1
a
5
xx  4 
x  1x  1
2)
5)
8)
11)
5
12 z
3)
16  15t
3
18t
a  5a  3
15 a 2
9u  1
16
6)
9)
12)
2m  3m  1
2
20
27 t 5
5
x4
9u 2  16u  25
12
4